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Stable Mappings and Their Singularities

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I: Preliminaries on Manifolds.- 1. Manifolds.- 2. Differentiable Mappings and Submanifolds.- 3. Tangent Spaces.- 4. Partitions of Unity.- 5. Vector Bundles.- 6. Integration of Vector Fields.- II: Transversality.- 1. Sard's Theorem.- 2. Jet Bundles.- 3. The Whitney C? Topology.- 4. Transversality.- 5. The Whitney Embedding Theorem.- 6. Morse Theory.- 7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- 1. Stable and Infinitesimally Stable Mappings.- 2. Examples.- 3. Immersions with Normal Crossings.- 4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- 1. The Weierstrass Preparation Theorem.- 2. The Malgrange Preparation Theorem.- 3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- 1. Another Formulation of Infinitesimal Stability.- 2. Stability Under Deformations.- 3. A Characterization of Trivial Deformations.- 4. Infinitesimal Stability => Stability.- 5. Local Transverse Stability.- 6. Transverse Stability.- 7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- 1. The Sr Classification.- 2. The Whitney Theory for Generic Mappings between 2-Manifolds.- 3. The Intrinsic Derivative.- 4. The Sr,s Singularities.- 5. The Thom-Boardman Stratification.- 6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- 1. Introduction.- 2. Finite Mappings.- 3. Contact Classes and Morin Singularities.- 4. Canonical Forms for Morin Singularities.- 5. Umbilics.- 6. Stable Mappings in Low Dimensions.- A. Lie Groups.- Symbol Index.

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